Algorithmica最新文献

We propose a new method based on discrete Fourier analysis to analyze the time evolutionary algorithms spend on plateaus. This immediately gives a concise proof of the classic estimate of the expected runtime of the ((1+1)) evolutionary algorithm on the Needle problem due to Garnier et al. (Evol Comput 7:173–203, 1999). We also use this method to analyze the runtime of the ((1+1)) evolutionary algorithm on a benchmark consisting of (n/ell ) plateaus of effective size (2^ell -1) which have to be optimized sequentially in a LeadingOnes fashion. Using our new method, we determine the precise expected runtime both for static and fitness-dependent mutation rates. We also determine the asymptotically optimal static and fitness-dependent mutation rates. For (ell = o(n)), the optimal static mutation rate is approximately 1.59/n. The optimal fitness dependent mutation rate, when the first k fitness-relevant bits have been found, is asymptotically (1/(k+1)). These results, so far only proven for the single-instance problem LeadingOnes, thus hold for a much broader class of problems. We expect similar extensions to be true for other important results on LeadingOnes. We are also optimistic that the Fourier analysis approach can be applied to other plateau problems as well.

The planar slope number ({{,textrm{psn},}}(G)) of a planar graph G is the minimum number of edge slopes in a planar straight-line drawing of G. It is known that ({{,textrm{psn},}}(G) in O(c^{Delta })) for every planar graph G of maximum degree (Delta ). This upper bound has been improved to (O(Delta ^5)) if G has treewidth three, and to (O(Delta )) if G has treewidth two. In this paper we prove ({{,textrm{psn},}}(G) le max {4,Delta }) when G is a Halin graph, and thus has treewidth three. Furthermore, we present the first polynomial upper bound on the planar slope number for a family of graphs having treewidth four. Namely we show that (O(Delta ^2)) slopes suffice for nested pseudotrees.

Based on well-known complexity theory conjectures, any polynomial-time kernelization algorithm for the NP-hard Line-Cover problem produces a kernel of size (Omega (k^2)), where k is the size of the sought line cover. Motivated by the current research in massive data processing, we study the existence of kernelization algorithms with limited space and time complexity for Line-Cover. We prove that every kernelization algorithm for Line-Cover takes time (Omega (n log k + k^2 log k)), and present a randomized kernelization algorithm for Line-Cover that produces a kernel of size bounded by (k^2), and runs in time ({mathcal {O}}(n log k + k^2 (log k log log k)^2)) and space ({mathcal {O}}(k^2log ^{2} k)). Our techniques are also useful for developing deterministic kernelization algorithms for Line-Cover with limited space and improved running time, and for developing streaming kernelization algorithms for Line-Cover with near-optimal update-time.

We study the Online Multiset Submodular Cover problem (OMSC), where we are given a universe U of elements and a collection of subsets (mathcal {S}subseteq 2^U). Each element (u_j in U) is associated with a nonnegative, nondecreasing, submodular polynomially computable set function (f_j). Initially, the elements are uncovered, and therefore we pay a penalty per each unit of uncovered element. Subsets with various coverage and cost arrive online. Upon arrival of a new subset, the online algorithm must decide how many copies of the arriving subset to add to the solution. This decision is irrevocable, in the sense that the algorithm will not be able to add more copies of this subset in the future. On the other hand, the algorithm can drop copies of a subset, but such copies cannot be retrieved later. The goal is to minimize the total cost of subsets taken plus penalties for uncovered elements. We present an (O(sqrt{rho _{max }}))-competitive algorithm for OMSC that does not dismiss subset copies that were taken into the solution, but relies on prior knowledge of the value of (rho _{max }), where (rho _{max }) is the maximum ratio, over all subsets, between the penalties covered by a subset and its cost. We provide an (Oleft( log (rho _{max }) sqrt{rho _{max }} right) )-competitive algorithm for OMSC that does not rely on advance knowledge of (rho _{max }) but uses dismissals of previously taken subsets. Finally, for the capacitated versions of the Online Multiset Multicover problem, we obtain an (O(sqrt{rho _{max }'}))-competitive algorithm when (rho _{max }') is known and an (Oleft( log (rho _{max }') sqrt{rho _{max }'} right) )-competitive algorithm when (rho _{max }') is unknown, where (rho _{max }') is the maximum ratio over all subset incarnations between the penalties covered by this incarnation and its cost.

Co-evolutionary algorithms have a wide range of applications, such as in hardware design, evolution of strategies for board games, and patching software bugs. However, these algorithms are poorly understood and applications are often limited by pathological behaviour, such as loss of gradient, relative over-generalisation, and mediocre objective stasis. It is an open challenge to develop a theory that can predict when co-evolutionary algorithms find solutions efficiently and reliable. This paper provides a first step in developing runtime analysis for population-based competitive co-evolutionary algorithms. We provide a mathematical framework for describing and reasoning about the performance of co-evolutionary processes. To illustrate the framework, we introduce a population-based co-evolutionary algorithm called PDCoEA, and prove that it obtains a solution to a bilinear maximin optimisation problem in expected polynomial time. Finally, we describe settings where PDCoEA needs exponential time with overwhelmingly high probability to obtain a solution.

Population diversity is crucial in evolutionary algorithms as it helps with global exploration and facilitates the use of crossover. Despite many runtime analyses showing advantages of population diversity, we have no clear picture of how diversity evolves over time. We study how the population diversity of ((mu +1)) algorithms, measured by the sum of pairwise Hamming distances, evolves in a fitness-neutral environment. We give an exact formula for the drift of population diversity and show that it is driven towards an equilibrium state. Moreover, we bound the expected time for getting close to the equilibrium state. We find that these dynamics, including the location of the equilibrium, are unaffected by surprisingly many algorithmic choices. All unbiased mutation operators with the same expected number of bit flips have the same effect on the expected diversity. Many crossover operators have no effect at all, including all binary unbiased, respectful operators. We review crossover operators from the literature and identify crossovers that are neutral towards the evolution of diversity and crossovers that are not.

Since the CSP dichotomy conjecture has been established, a number of other dichotomy questions have attracted interest, including one for list homomorphism problems of signed graphs. Signed graphs arise naturally in many contexts, including for instance nowhere-zero flows for graphs embedded in non-orientable surfaces. The dichotomy classification is known for homomorphisms without list restrictions, so it is surprising that it is not known, or even conjectured, if lists are present since this usually makes the classifications easier to obtain. There is however a conjectured classification, due to Kim and Siggers, in the special case of “semi-balanced” signed graphs. These authors confirmed their conjecture for the class of reflexive signed graphs. As our main result we verify the conjecture for irreflexive signed graphs. For this purpose, we prove an extension result for two-directional ray graphs which is of independent interest and which leads to an analogous extension result for interval graphs. Moreover, we offer an alternative proof for the class of reflexive signed graphs, and a direct polynomial-time algorithm in the polynomial cases where the previous algorithms used algebraic methods of general CSP dichotomy theorems. For both reflexive and irreflexive cases the dichotomy classification depends on a result linking the absence of certain structures to the existence of a special ordering. The structures are used to prove the NP-completeness and the ordering is used to design polynomial algorithms.

A conflict-free coloring of a graph G is a (partial) coloring of its vertices such that every vertex u has a neighbor whose assigned color is unique in the neighborhood of u. There are two variants of this coloring, one defined using the open neighborhood and one using the closed neighborhood. For both variants, we study the problem of deciding whether the conflict-free coloring of a given graph G is at most a given number k.

In this work, we investigate the relation of clique-width and minimum number of colors needed (for both variants) and show that these parameters do not bound one another. Moreover, we consider specific graph classes, particularly graphs of bounded clique-width and types of intersection graphs, such as distance hereditary graphs, interval graphs and unit square and disk graphs. We also consider Kneser graphs and split graphs. We give (often tight) upper and lower bounds and determine the complexity of the decision problem on these graph classes, which improve some of the results from the literature. Particularly, we settle the number of colors needed for an interval graph to be conflict-free colored under the open neighborhood model, which was posed as an open problem.

Motivated by applications in DNA-nanotechnology, theoretical investigations in algorithmic tile-assembly have blossomed into a mature theory. In addition to computational universality, the abstract Tile Assembly Model (aTAM) was shown to be intrinsically universal (FOCS 2012), a strong notion of completeness where a single tile set is capable of simulating the full dynamics of all systems within the model; however, this construction fundamentally required non-deterministic tile attachments. This was confirmed necessary when it was shown that the class of directed aTAM systems, those where all possible sequences of tile attachments result in the same terminal assembly, is not intrinsically universal (FOCS 2016). Furthermore, it was shown that the non-cooperative aTAM, where tiles only need to match on 1 side to bind rather than 2 or more, is not intrinsically universal (SODA 2014) nor computationally universal (STOC 2017). Building on these results to further investigate the other dynamics, Hader et al. examined several tile-assembly models which varied across (1) the numbers of dimensions used, (2) how tiles diffused through space, and (3) whether each system is directed, and determined which models exhibited intrinsic universality (SODA 2020). In this paper we extend those results to provide direct comparisons of the various models against each other by considering intrinsic simulations between models. Our results show that in some cases, one model is strictly more powerful than another, and in others, pairs of models have mutually exclusive capabilities. This paper is a greatly expanded version of that which appeared in ICALP 2023.

A permutation (pi ) is a merge of a permutation (sigma ) and a permutation (tau ), if we can color the elements of (pi ) red and blue so that the red elements have the same relative order as (sigma ) and the blue ones as (tau ). We consider, for fixed hereditary permutation classes (mathcal {C}) and (mathcal {D}), the complexity of determining whether a given permutation (pi ) is a merge of an element of (mathcal {C}) with an element of (mathcal {D}). We develop general algorithmic approaches for identifying polynomially tractable cases of merge recognition. Our tools include a version of streaming recognizability of permutations via polynomially constructible nondeterministic automata, as well as a concept of bounded width decomposition, inspired by the work of Ahal and Rabinovich. As a consequence of the general results, we can provide nontrivial examples of tractable permutation merges involving commonly studied permutation classes, such as the class of layered permutations, the class of separable permutations, or the class of permutations avoiding a decreasing sequence of a given length. On the negative side, we obtain a general hardness result which implies, for example, that it is NP-complete to recognize the permutations that can be merged from two subpermutations avoiding the pattern 2413.